On two generalizations of semi-projective modules: SGQ-projective and $pi$-semi-projective

Document Type: Research Paper


Quchan university of Advanced Technology


Let $R$ be a ring and $M$ a right $R$-module with $S=End_R(M)$. A module $M$ is called semi-projective if for any epimorphism $f:M\rightarrow N$, where $N$ is a submodule of $M$, and for any homomorphism $g: M\rightarrow N$, there exists $h:M\rightarrow M$ such that $fh=g$. In this paper, we study SGQ-projective and $\pi$-semi-projective modules as two generalizations of semi-projective modules. A module $M$ is called an SGQ-projective module if for any $\phi\in S$, there exists a right ideal $X_\phi$ of $S$ such that $D_S(\Im \phi)=\phi S\oplus X_\phi$ as right $S$-modules. We call $M$ a $\pi$-semi-projective module if for any $0\neq s\in S$, there exists a positive integer $n$ such that $s^n\neq 0$ and any $R$-homomorphism from $M$ to $s^nM$ can be extended to an endomorphism of $M$. Some properties of this class of modules are investigated.